Recall that, for an integral variety \(X\)  over a field \(K\) , a subset \(S\subseteq ~X(K)\)  is called thin (in \(X\)  or in \(X(K)\) ) if there exist an integer \(n\) , integral \(K\) -varieties \(Y_1,\ldots ,Y_n\)  and dominant generically finite separable morphisms \(\pi _i\colon Y_i\rightarrow X\)  of degree at least two such that \(S\setminus \bigcup _{i=1}^n \pi _i(Y_i(K))\)  is not dense in \(X\) . One says that \(X\)  satisfies the Hilbert property over \(K\) if \(X(K)\)  is not thin.
The Hilbert property is closely related to the inverse Galois problem for \(\mathbb {Q}\)  (see [1]}) and studied in
[2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}.
